Perturbation theory in $U$

A problem with perturbation theory here is the fact that the interaction $U$ between the two electrons is not small.

The unperturbed states $\vert\alpha\rangle$ and $\vert\beta\rangle$ for the orbital wave functions (cf. section III.2.3.2) are the eigenstates of the hydrogen problem, Eq. (II.1.20),

$$\vert\alpha\rangle = \vert nlm\rangle,\quad \vert\beta\rangle = \vert n'l'm'\rangle.$$ (3.2)

Note that these do not contain a spin index.